Olympiad problems

2020 Brazil Undergrad MO, Problem 6

Tags: function , college contests , quadratics , roots , Fixed point , Brazilian Undergrad MO , Brazilian Undergrad MO 2020

Let $f(x) = 2x^2 + x - 1, f^{0}(x) = x$, and $f^{n+1}(x) = f(f^{n}(x))$ for all real $x>0$ and $n \ge 0$ integer (that is, $f^{n}$ is $f$ iterated $n$ times). a) Find the number of distinct real roots of the equation $f^{3}(x) = x$ b) Find, for each $n \ge 0$ integer, the number of distinct real solutions of the equation $f^{n}(x) = 0$

2018 Brazil Undergrad MO, 7

Tags: graph theory , isomorphism , Brazilian Undergrad MO , superior algebra

Unless of isomorphisms, how many simple four-vertex graphs are there?

2020 Brazil Undergrad MO, Problem 1

Tags: limits , calculus , Brazilian Undergrad MO , Brazilian Undergrad MO 2020 , geometry

Let $R > 0$, be an integer, and let $n(R)$ be the number um triples $(x, y, z) \in \mathbb{Z}^3$ such that $2x^2+3y^2+5z^2 = R$. What is the value of $\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?

2018 Brazil Undergrad MO, 22

Tags: Brazilian Undergrad MO , college contest , calculus , integration

What is the value of the improper integral $ \int_0 ^ {\pi} \log (\sin (x)) dx$?

2019 Brazil Undergrad MO, 3

Tags: calculus , system of equations , college contests , algebra , Brazilian Undergrad MO , Brazilian Undergrad MO 2019

Let $a,b,c$ be constants and $a,b,c$ are positive real numbers. Prove that the equations $2x+y+z=\sqrt{c^2+z^2}+\sqrt{c^2+y^2}$ $x+2y+z=\sqrt{b^2+x^2}+\sqrt{b^2+z^2}$ $x+y+2z=\sqrt{a^2+x^2}+\sqrt{a^2+y^2}$ have exactly one real solution $(x,y,z)$ with $x,y,z \geq 0$.

2020 Brazil Undergrad MO, Problem 3

Tags: linear algebra , matrix , Brazilian Undergrad MO , Brazilian Undergrad MO 2020 , number theory

Let $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$ be the finite field with $13$ elements (with sum and product modulus $13$). Find how many matrix $A$ of size $5$ x $5$ with entries in $\mathbb{F}_{13}$ exist such that $$A^5 = I$$ where $I$ is the identity matrix of order $5$

2020 Brazil Undergrad MO, Problem 2

Tags: Brazilian Undergrad MO 2020 , number theory , Sequences , Brazilian Undergrad MO

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is fibonatic when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not fibonatic integers.

2018 Brazil Undergrad MO, 10

Tags: Brazilian Undergrad MO , college contest , limits , calculus

How many ordered pairs of real numbers $ (a, b) $ satisfy equality $\lim_{x \to 0} \frac{\sin^2x}{e^{ax}-2bx-1}= \frac{1}{2}$?

2018 Brazil Undergrad MO, 19

Tags: complex numbers , calculus , Brazilian Undergrad MO

What is the largest amount of complex $ z $ solutions a system can have? $ | z-1 || z + 1 | = 1 $ $ Im (z) = b? $ (where $ b $ is a real constant)

2017 Brazil Undergrad MO, 2

Tags: number theory , prime numbers , Sequence , Brazilian Undergrad MO

Let $a$ and $b$ be fixed positive integers. Show that the set of primes that divide at least one of the terms of the sequence $a_n = a \cdot 2017^n + b \cdot 2016^n$ is infinite.

2008 Brazil Undergrad MO, 3

Tags: search , real analysis , real analysis unsolved , Brazilian Undergrad MO

Prove that there are real numbers $ a_1, a_2, ..$ such that: i) For all real numbers x, the serie $ f(x) \equal{} \sum_{n \equal{} 1}^\infty a_nx^n$ converge; ii) f is a bijection of R to R; iii) f'(x) >0; iv) f(Q) = A, where Q is the set of rational numbers and A is the set of algebraic numbers.

2018 Brazil Undergrad MO, 24

Tags: Summation , Brazilian Undergrad MO , superior algebra

What is the value of the series $\sum_{1 \leq l <m<n} \frac{1}{5^l3^m2^n}$

2018 Brazil Undergrad MO, 25

Tags: topology , Brazilian Undergrad MO , group theory

Consider the $ \mathbb {Z} / (10) $ additive group automorphism group of integers module $10$, that is, $ A = \left \{\phi: \mathbb {Z} / (10) \to \mathbb {Z} / (10) | \phi-automorphism \right \}$

2018 Brazil Undergrad MO, 4

Tags: Sets , college contests , Brazilian Undergrad MO

Consider the property that each a element of a group $G$ satisfies $a ^ 2 = e$, where e is the identity element of the group. Which of the following statements is not always valid for a group $G$ with this property? (a) $G$ is commutative (b) $G$ has infinite or even order (c) $G$ is Noetherian (d) $G$ is vector space over $\mathbb{Z}_2$

2021 Brazil Undergrad MO, Problem 1

Tags: Brazilian Undergrad MO , Matrix algebra , Square matrix , linear algebra , Brazilian Undergrad MO 2021

Consider the matrices like $$M= \left( \begin{array}{ccc} a & b & c \\ c & a & b \\ b & c & a \end{array} \right)$$ such that $det(M) = 1$. Show that a) There are infinitely many matrices like above with $a,b,c \in \mathbb{Q}$ b) There are finitely many matrices like above with $a,b,c \in \mathbb{Z}$

2019 Brazil Undergrad MO, 1

Tags: college contests , Matrices , linear algebra , polynomial , Brazilian Undergrad MO 2019 , Brazilian Undergrad MO

Let $ I $ and $ 0 $ be the square identity and null matrices, both of size $ 2019 $. There is a square matrix $A$ with rational entries and size $ 2019 $ such that: a) $ A ^ 3 + 6A ^ 2-2I = 0 $? b) $ A ^ 4 + 6A ^ 3-2I = 0 $?

2019 Brazil Undergrad MO, Problem 5

Tags: Brazilian Undergrad MO 2019 , Brazilian Undergrad MO , number theory , combinatorics , counting , limit

Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$